Hedging with Credibility When Assets Can Jump

Abstract

Hedging options in non-Gaussian models is a well-known and difficult task, yet remaining important for risk practitioners from banks to insurance companies. Hence, solutions through the quadratic hedging methods have been recently suggested, see Cont and Tankov (2004), Riesner (2006) and Vandaele and Vanmaele (2008). Although their suggested ratios to invest in the underlying asset for an optimal replication are different from each to other, they, however, share a common structure which makes their implementation non obvious. This structure originates in the integral part of the partial integro-differential equation and stems from the expectation of option prices taken over the random jump sizes. Although non-straightforward numerical integrations can be used to implement this quantity, they have to be modified and adapted to suit the choice of the random jump size distributions, resulting in a cumbersome task. Hence, implementation efficiency has still to be addressed. Using a locally risk-minimizing hedging strategy together with an elegant result of Hille and Phillips (1957), this paper shows how to efficiently compute the expectation of the option prices taken over the random jump sizes of any Levy processes, be they of the finite or of the infinite activity. Hence, all the optimal ratios suggested by the aforementioned authors can be evaluated by adding a minor factor to a fast Fourier transform pricing formula and thereby gaining its computation efficiency.